Polygonal Domains Research Articles

Bernstein inequality on conic domains and triangles

We establish weighted Bernstein inequalities in Lp space for the doubling weight on the conic surface V0d+1={(x,t):‖x‖=t,x∈Rd,t∈[0,1]} as well as on the solid cone bounded by the conic surface and the hyperplane t=1, which becomes a triangle on the plane when d=1. While the inequalities for the derivatives in the t variable behave as expected, there are inequalities for the derivatives in the x variables that are stronger than what one may have expected. As an example, on the triangle {(x1,x2):x1≥0,x2≥0,x1+x2≤1}, the usual Bernstein inequality for the derivative ∂1 states that ‖ϕ1∂1f‖p,w≤cn‖f‖p,w with ϕ1(x1,x2)≔x1(1−x1−x2), whereas our new result gives ‖(1−x2)−1/2ϕ1∂1f‖p,w≤cn‖f‖p,w.The new inequality is stronger and points out a phenomenon unobserved hitherto for polygonal domains.

Theoretical analysis and construction of numerical method for solving the Navier–Stokes equations in rotation form with corner singularity

In this paper, we introduced the notion of an Rν-generalized solution of the Oseen problem in rotation form in two-dimensional polygonal domain with a reentrant corner at the boundary. Its existence and uniqueness in the nonsymmetric variational formulation of the problem is established. An estimate related to the conservation of the energy balance of the approximation velocity field for the Navier–Stokes problem in the rotation form is obtained. A weighted finite element method is constructed. A series of numerical experiments of test examples based on the proposed method is carried out. A comparative analysis of the approach with the classical FEM is performed. The results of numerical experiments showed a significant advantage of the proposed approach.

A modified stable node-based smoothed finite element method based on low-quality unstructured mesh

A modified stable node-based smoothed finite element method (M-SNS-FEM) based on linear low-quality unstructured mesh is proposed for two-dimensional elastic static analysis and free vibration analysis. Firstly, the weakened weak formulation based on nodes is considered under finite element background, framing the so-called node-based domains. Secondly, the polygonal smoothing domain around each node is approximated as ellipse for low-quality unstructured mesh, and the locations of auxiliary integration points are further chosen. And then, the integration scheme of M-SNS-FEM is obtained by simplifying the discrete equations acquired by auxiliary integration points. Finally, various problems containing both benchmark cases and practical engineering examples are studied, and the applicability and accuracy of M-SNS-FEM in linear low-quality unstructured mesh can be verified. The results show that the proposed approach can always achieve high accuracy as the mesh quality deteriorates gradually and local mesh deforms sharply.

A Symmetric Interior Penalty Method for an Elliptic Distributed Optimal Control Problem with Pointwise State Constraints

Abstract We construct a symmetric interior penalty method for an elliptic distributed optimal control problem with pointwise state constraints on general polygonal domains. The resulting discrete problems are quadratic programs with simple box constraints that can be solved efficiently by a primal-dual active set algorithm. Both theoretical analysis and corroborating numerical results are presented.

Shape prediction on the basis of spectrum using neural networks

We have developed a deep neural network that reconstructs the shape of a polygonal domain given the first hundred of its Dirichlet Laplacian eigenvalues. Having an encoder-decoder structure, the network maps input spectra to a latent space and then predicts the discretized image of the domain on a square grid. Tested on randomly generated pentagons, the predictions prove to be highly accurate for both concave and convex pentagons. Our analysis indicates that the network has discovered fundamental properties of the Laplacian operator, the scaling rule, and the continuous rotational symmetry. Additionally, the latent variables are strongly correlated with Weyl's parameters (area, perimeter, and a certain function of the angles) of the test polygons.

Global C2,α estimates for the Monge-Ampère equation on polygonal domains in the plane

We classify global solutions of the Monge-Amp\\`ere equation $\\det D^2 u=1$ on the first quadrant in the plane with quadratic boundary data. As an application, we obtain global $C^\{2,\\alpha\}$ estimates for the non-degenerate Monge-Amp\\`ere equation in convex polygonal domains in $\\Bbb\{R\}^2$ provided a globally $C^2$, convex strict subsolution exists.

The coefficients in the asymptotic expansion of solutions of second‐order hyperbolic problems in polygonal domains

Solutions of boundary value problems for linear partial differential equations are known to exhibit singular behaviors near the boundary of nonsmooth domains. In this case, one is usually interested on the asymptotic behavior of the solutions near the geometric singularities. However, for both mathematical and engineering purposes, it is important to have extraction formulas for the coefficients in the asymptotic expansion. In this paper, we consider initial boundary value problems for the wave equation in two‐dimensional domains with corners and derive, by means of Fourier analysis, explicit computable expressions for the coefficients in the asymptotic expansion of the solution near the corners. The explicit structure of the singular solutions presented herein makes it fairly easy to compute numerically. Thus, the formulas can be used to construct postprocessing procedures for improving the accuracy of standard numerical techniques of approximating the solution and for improving the rates of convergence in the error estimates. The analysis presented herein would also apply to more general hyperbolic or parabolic equations with constant coefficients.

Analysis of an HDG method for the Navier–Stokes equations with Dirac measures

In two dimensions, we analyze a hybridized discontinuous Galerkin (HDG) method for the Navier–Stokes equations with Dirac measures. The approximate velocity field obtained by the HDG method is shown to be pointwise divergence-free and H(div)-conforming. Under a smallness assumption on the continuous and discrete solutions, a posteriori error estimator, that provides an upper bound for the L2-norm in the velocity, is proposed in the convex domain. In the polygonal domain, reliable and efficient a posteriori error estimator for the W1,q-seminorm in the velocity and Lq-norm in the pressure is also proved. Finally, a Banach’s fixed point iteration method and an adaptive HDG algorithm are introduced to solve the discrete system and show the performance of the obtained a posteriori error estimators.

Energy minimizing solutions to slightly subcritical elliptic problems on nonconvex polygonal domains

<abstract><p>In this paper we are concerned with the Lane-Emden-Fowler equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{\begin{array}{rll}-\Delta u & = u^{\frac{n+2}{n-2}- \varepsilon}& {\rm{in}}\; \Omega, \\ u&>0& {\rm{in}}\; \Omega, \\ u& = 0& {\rm{on}}\; \partial \Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \Omega \subset \mathbb{R}^n $ ($ n \geq 3 $) is a nonconvex polygonal domain and $ \varepsilon > 0 $. We study the asymptotic behavior of minimal energy solutions as $ \varepsilon > 0 $ goes to zero. A main part is to show that the solution is uniformly bounded near the boundary with respect to $ \varepsilon > 0 $. The moving plane method is difficult to apply for the nonconvex polygonal domain. To get around this difficulty, we derive a contradiction after assuming that the solution blows up near the boundary by using the Pohozaev identity and the Green's function.</p></abstract>

A Morley-type virtual element approximation for a wind-driven ocean circulation model on polygonal meshes

In this work, we propose and analyze a Morley-type virtual element method to approximate the Stommel–Munk model in stream-function form. The discretization is based on the fully nonconforming virtual element approach presented in Antonietti et al., (2018) and Zhao et al., (2018). The analysis restricts to simply connected polygonal domains, not necessarily convex. Under standard assumptions on the computational domain we derive some inverse estimates, norm equivalence and approximation properties for an enriching operatorEh defined from the nonconforming space into its H2-conforming counterpart. With the help of these tools we prove optimal error estimates for the stream-function in broken H2-, H1- and L2-norms under minimal regularity condition on the weak solution. Employing postprocessing formulas and adequate polynomial projections we compute from the discrete stream-function further fields of interest, such as: the velocity and vorticity. Moreover, for these postprocessed variables we establish error estimates. Finally, we report practical numerical experiments on different families of polygonal meshes.

Mathematical Model and Solution Algorithm for Virtual Localization Problem

Introduction. The optimization placement problem refereed to virtual localization is studied. This problem is motivated by the need to optimize the production of parts from near-net shape blanks using CNC machines. The known algorithms for solving the virtual localization problem come down to determining the location parameters of the part CAD model inside the point cloud obtained by scanning the workpiece surface. The main disadvantage of such algorithms is the use of criteria that are insensitive to the intersection of the surfaces of the part and the workpiece. In order to prevent such errors in production conditions, it is necessary to involve a human operator in conducting operations based on virtual localization. In this way, the virtual localization problem of complex shape objects is of paramount importance. The purpose of the paper is to propose a new approach for solving the virtual localization problem. Results. A new mathematical model of the virtual localization problem based on the phi-function technique is proposed. We developed a solution strategy that combines algorithm of generating feasible starting points with non-linear optimization procedure. The testing of the proposed approach was carried out for a two-dimensional case. The computational results illustrated with graphical illustrations are provided that show the efficiency of the proposed algorithm. Conclusions. The obtained results show that the use of the phi-functions technique prevents the occurrence of erroneous solutions with the intersection of the workpiece surfaces. An algorithm for solving the problem of virtual localization in a two-dimensional formulation for the case when the part and the workpiece are convex polygons has been developed. For the considered test problems, the solution time did not exceed 2.5 sec, which fully meets the requirements of industrial use. In the future, it is planned to extend the proposed method to the cases when the CAD model of the part has an arbitrary shape and is formed by Boolean operations on geometric primitives. Keywords: polygonal domain, phi-function technique, virtual localization, CNC machining.

Hp-FEM for reaction–diffusion equations. II: robust exponential convergence for multiple length scales in corner domains

Abstract In bounded, polygonal domains $\varOmega \subset {{\mathbb {R}}}^2$ with Lipschitz boundary $\partial \varOmega $ consisting of a finite number of Jordan curves admitting analytic parametrizations, we analyze $hp$-FEM discretizations of linear, second-order, singularly perturbed reaction–diffusion equations on so-called geometric boundary layer meshes. We prove, under suitable analyticity assumptions on the data, that these $hp$-FEM afford exponential convergence in the natural ‘energy’ norm of the problem, as long as the geometric boundary layer mesh can resolve the smallest length scale present in the problem. Numerical experiments confirm the robust exponential convergence of the proposed $hp$-FEM.

A rational-expansion-based method to compute Gabor coefficients of 2D indicator functions supported on polygonal domain

We propose a method to compute Gabor coefficients of a two-dimensional (2D) indicator function supported on a polygonal domain by means of rational expansion of the Faddeeva function and by solving second-order linear difference equations. This method has the following three attractive features: (1) the problem of computing Gabor coefficients is formulated as the calculation of a sequence of integrals with a uniform structure, (2) a rational expansion based on fast Fourier transform (FFT) is used to approximate the Faddeeva function on the entire complex plane, (3) second-order inhomogeneous linear difference equations are derived for previous integrals and they are solved stably with Olver’s algorithm. Numerical quadrature to compute Gabor coefficients is avoided. Numerical examples show this rational-expansion-based method significantly outperforms numerical quadrature in terms of computation time while maintaining accuracy.

Exponential convergence of hp FEM for spectral fractional diffusion in polygons

For the spectral fractional diffusion operator of order 2s, s in (0,1), in bounded, curvilinear polygonal domains varOmega subset {{mathbb {R}}}^2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm {mathbb {H}}^s(varOmega ). The first hp discretization is based on writing the solution as a co-normal derivative of a 2+1-dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations invarOmega . Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in varOmega , exponential convergence rates for solutions uin {mathbb {H}}^s(varOmega ) of mathcal {L}^s u = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towardspartial varOmega . The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of mathcal {L}^{-s} combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations invarOmega . The present analysis for either approach extends to (polygonal subsets {widetilde{mathcal {M}}} of) analytic, compact 2-manifolds {mathcal {M}}, parametrized by a global, analytic chart chi with polygonal Euclidean parameter domain varOmega subset {{mathbb {R}}}^2. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced.

A [formula omitted]-nonconforming virtual element methods for the vibration and buckling problems of thin plates

In this work, we study the C0-nonconforming VEM for the fourth-order eigenvalue problems modeling the vibration and buckling problems of thin plates with clamped boundary conditions on general shaped polygonal domain, possibly even nonconvex domain. By employing the enriching operator, we have derived the convergence analysis in discrete H2 seminorm, and H1, L2 norms for both problems. We use the Babuška–Osborn spectral theory (Babuška and Osborn, 1991), to show that the introduced schemes provide well approximation of the spectrum and prove optimal order of rate of convergence for eigenfunctions and double order of rate of convergence for eigenvalues. Finally, numerical results are presented to show the good performance of the method on different polygonal meshes.

Finite Element Approximations for PDEs with Irregular Dirichlet Boundary Data on Boundary Concentrated Meshes

Abstract This paper is concerned with finite element error estimates for second order elliptic PDEs with inhomogeneous Dirichlet boundary data in convex polygonal domains. The Dirichlet boundary data are assumed to be irregular such that the solution of the PDE does not belong to H 2 ⁢ ( Ω ) {H^{2}(\Omega)} but only to H r ⁢ ( Ω ) {H^{r}(\Omega)} for some r ∈ ( 1 , 2 ) {r\in(1,2)} . As a consequence, a discretization of the PDE with linear finite elements exhibits a reduced convergence rate in L 2 ⁢ ( Ω ) {L^{2}(\Omega)} and H 1 ⁢ ( Ω ) {H^{1}(\Omega)} . In order to restore the best possible convergence rate we propose and analyze in detail the usage of boundary concentrated meshes. These meshes are gradually refined towards the whole boundary. The corresponding grading parameter does not only depend on the regularity of the Dirichlet boundary data and their discrete implementation but also on the norm, which is used to measure the error. In numerical experiments we confirm our theoretical results.

Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data

First-order convergence in time and space is proved for a fully discrete semi-implicit finite element method for the two-dimensional Navier–Stokes equations with L2 initial data in convex polygonal domains, without extra regularity assumptions or grid-ratio conditions. The proof utilises the smoothing properties of the Navier–Stokes equations in the analysis of the consistency errors, an appropriate duality argument, and the smallness of the numerical solution in the discrete L2(0, tm; H1) norm when tm is smaller than some constant. Numerical examples are provided to support the theoretical analysis.

Finite element approximation of invariant manifolds by the parameterization method

We combine the parameterization method for invariant manifolds with the finite element method for elliptic PDEs, to obtain a new computational framework for high order approximation of invariant manifolds attached to unstable equilibrium solutions of nonlinear parabolic PDEs. The parameterization method provides an infinitesimal invariance equation for the invariant manifold, which we solve via a power series ansatz. A power matching argument leads to a recursive systems of linear elliptic PDEs—the so called homological equations—whose solutions are the power series coefficients of the parameterization. The homological equations are solved recursively to any desired order N using finite element approximation. The end result is an N-th order polynomial approximation of a chart map of the manifold, with coefficients in an appropriate finite element space. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two dimensional polygonal domains (not necessary simply connected), for equilibrium solutions with Morse indices one and two. We implement a-posteriori error indicators which provide numerical evidence in support of the claim that the manifolds are computed accurately.

Quantitative bounds on impedance-to-impedance operators with applications to fast direct solvers for PDEs

We prove quantitative norm bounds for a family of operators involving impedance boundary conditions on convex, polygonal domains. A robust numerical construction of Helmholtz scattering solutions in variable media via the Dirichlet-to-Neumann operator involves a decomposition of the domain into a sequence of rectangles of varying scales and constructing impedance-to-impedance boundary operators on each subdomain. Our estimates in particular ensure the invertibility, with quantitative bounds in the frequency, of the merge operators required to reconstruct the original Dirichlet-to-Neumann operator in terms of these impedance-to-impedance operators of the sub-domains. A key step in our proof is to obtain Neumann and Dirichlet boundary trace estimates on solutions of the impedance problem, which are of independent interest. In addition to the variable media setting, we also construct bounds for similar merge operators in the obstacle scattering problem.

Clique-Based Separators for Geometric Intersection Graphs

Let F be a set of n objects in the plane and let mathcal {G}^{times }(F) be its intersection graph. A balanced clique-based separator of mathcal {G}^{times }(F) is a set mathcal {mathcal {S}} consisting of cliques whose removal partitions mathcal {G}^{times }(F) into components of size at most delta n, for some fixed constant delta <1. The weight of a clique-based separator is defined as sum _{Cin mathcal {mathcal {S}}}log (|C|+1). Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 2020) proved that if S consists of convex fat objects, then mathcal {G}^{times }(F) admits a balanced clique-based separator of weight O(sqrt{n}). We extend this result in several directions, obtaining the following results. (i) Map graphs admit a balanced clique-based separator of weight O(sqrt{n}), which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3}log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(sqrt{n}log n). (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3}log n). (iv) Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(sqrt{n}+rlog (n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for Maximum Independent Set (and, hence, Vertex Cover), for Feedback Vertex Set, and for q-Coloring for constant q in these graph classes.